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Counting points on hyperelliptic curves over finite fields
International audienceWe describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor's division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature
Curves, dynamical systems and weighted point counting
Suppose X is a (smooth projective irreducible algebraic) curve over a finite
field k. Counting the number of points on X over all finite field extensions of
k will not determine the curve uniquely. Actually, a famous theorem of Tate
implies that two such curves over k have the same zeta function (i.e., the same
number of points over all extensions of k) if and only if their corresponding
Jacobians are isogenous. We remedy this situation by showing that if, instead
of just the zeta function, all Dirichlet L-series of the two curves are equal
via an isomorphism of their Dirichlet character groups, then the curves are
isomorphic up to "Frobenius twists", i.e., up to automorphisms of the ground
field. Since L-series count points on a curve in a "weighted" way, we see that
weighted point counting determines a curve. In a sense, the result solves the
analogue of the isospectrality problem for curves over finite fields (also know
as the "arithmetic equivalence problem"): it says that a curve is determined by
"spectral" data, namely, eigenvalues of the Frobenius operator of k acting on
the cohomology groups of all l-adic sheaves corresponding to Dirichlet
characters. The method of proof is to shown that this is equivalent to the
respective class field theories of the curves being isomorphic as dynamical
systems, in a sense that we make precise.Comment: 11 page
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